Short Course Spectral-element solution of the elastic wave equation Andreas Fichtner

1. Short Course Spectral-element solution of the elastic wave equation Andreas Fichtner

2. OUTLOOK • 1. Introduction • recalling the elastic wave equation • 2. The spectral-element method: General concept • domain mapping • from space-continuous to space-discrete • time interpolation • Gauss-Lobatto-Legendre interpolation and integration • 3. A special flavour of the spectral-element method: SES3D • programme code description • computation of synthetic seismograms • long-wavelength equivalent models • 4. Computation of sensitivity kernels using SES3D • construction of the adjoint source • 5. On the sensitive nature of numerical methods: Calls to caution!

3. Introduction Recalling the elastic wave equation

4. THE ELASTIC WAVE EQUATION Elastic wave equation: Subsidiary conditions: • No analytical solutions exist for Earth models with 3D heterogeneities. • We use numerical methods: • - Finite-difference method (FDM) • - Finite-element method (FEM) • - Direct-solution method (DSM) • - Discontinuous Galerkin method (DGM) • - … • - Spectral-element method

5. The spectral-element method General concept

6. SPECTRAL-ELEMENT METHOD: General Concept see Bernhard‘s web site: www.geophysik.uni-muenchen.de/~bernhard

7. SPECTRAL-ELEMENT METHOD: General Concept Subdivision of the computational domain into hexahedral elements: (a) 2D subdivision that honours layer boundaries (b) Subdivision of the globe (cubed sphere) (c) Subdivision with topography

8. SPECTRAL-ELEMENT METHOD: General Concept Mapping to the unit cube:

9. SPECTRAL-ELEMENT METHOD: General Concept Representation in terms of polynomials: (within the unit interval [-1 1]) Nth-degree Lagrange polynomials → We can transform the partial differential equation into an ordinary differential equation where we solve for the polynomial coefficients: mass matrix stiffness matrix

10. SPECTRAL-ELEMENT METHOD: General Concept Choice of the collocation points: Interpolation of Runge‘s function R(x) using 6th-order polynomials and equidistant collocation points interpolant Runge‘s phenomenon

11. SPECTRAL-ELEMENT METHOD: General Concept Choice of the collocation points: Interpolation of Runge‘s function R(x) using 6th-order polynomials and Gauss-Lobatto-Legendre collocation points [ roots of (1-x2)LoN-1= completed Lobatto polynomial ] interpolant We should use the GLL points as collocation points for the Lagrange polynomials.

12. SPECTRAL-ELEMENT METHOD: General Concept • Example: GLL Lagrange polynomials of degree 6 • collocation points = GLL points • global maxima at the collocation points

13. SPECTRAL-ELEMENT METHOD: General Concept Numerical quadrature to determine mass and stiffness matrices: Quadrature node points = GLL points → The mass matrix is diagonal, i.e., trivial to invert. → This is THE advantage of the spectral-element method. Time extrapolation: … this can be solved on a computer.

14. A special flavour of the spectral-element method SES3D

15. SES3D: General Concept • Simulation of elastic wave propagation in a spherical section. • Spectral-element discretisation. • Computation of Fréchet kernels using the adjoint method. • Operates in natural spherical coordinates! • 3D heterogeneous, radially anisotropic, • visco-elastic. • PML as absorbing boundaries. • Programme philosophy: • Puritanism [easy to modify and • adapt to different problems, easy • implementation of 3D models, • simple code]

16. SES3D: Example Southern Greece 8 June, 2008 Mw=6.3

17. SES3D: Example Southern Greece 8 June, 2008 Mw=6.3 • Input files [geometric setup, source, receivers, Earth model] • Forward simulation [wavefield snapshots and seismograms] • Adjoint simulation [adjoint source, Fréchet kernels]

18. SES3D: Input files • Par: • Numerical simulation parameters • Geometrical setup • Seismic Source • Parallelisation • stf: • Source time function • recfile: • - Receiver positions

19. SES3D: Parallelisation • Spherical section subdivided into equal-sized subsections • Each subsection is assigned to one processor. • Communication: MPI

20. SES3D: Source time function Source time function - time step and length agree with the simulation parameters - PMLs work best with bandpass filtered source time functions - Example: bandpass [50 s to 200 s]

21. LONG WAVELENGTH EQUIVALENT MODELS Single-layered crust that coincides with the upper layer of elements … … and PREM below boundary between the upper 2 layers of elements lon=142.74° lat=-5.99° d=80 km vertical displacement SA08 lon=150.89° lat=-25.89° Dalkolmo & Friederich, 1995. Complete synthetic seismograms for a spherically symmetric Earth …, GJI, 122, 537-550

22. LONG WAVELENGTH EQUIVALENT MODELS verification 2-layered crustthat does not coincide with a layer of elements … … and PREM below boundary between the upper 2 layers of elements lon=142.74° lat=-5.99° d=80 km vertical displacement SA08 lon=150.89° lat=-25.89° Dalkolmo & Friederich, 1995. Complete synthetic seismograms for a spherically symmetric Earth …, GJI, 122, 537-550

23. LONG WAVELENGTH EQUIVALENT MODELS long wavelength equivalent models • Replace original crustral • model by a long- • wavelength equivalent • model … • … which is transversely • isotropic [Backus, 1962]. • The optimal smooth model • is found by dispersion • curve matching. Fichtner & Igel, 2008. Efficient numerical surface wave propagation through the optimisation of discrete crustal models, GJI.

24. LONG WAVELENGTH EQUIVALENT MODELS long wavelength equivalent models Minimisation of the phase velocity differences for the fundamental and higher modes in the frequency range of interest through simulated annealing.

25. LONG WAVELENGTH EQUIVALENT MODELS long wavelength equivalent models crustal thickness map (crust2.0) • 3D solution: interpolation of long wavelength equivalent profiles to obtain 3D crustal model. • Problem 1: crustal structure not well constrained (receiver function non-uniqueness) • Problem 2: abrupt changes in crustal structure (not captured by pointwise RF studies)

26. Computation of Fréchet kernels

27. SES3D: Computation of Fréchet kernels Vertical component velocity seismogram: taper target waveform divide by L2 norm invert time axis Adjoint source time function: [cross-correlation time shift]

28. On the sensitive nature of numerical methods Calls to caution!

29. SES3D: Calls to caution! • Long-term instability of PMLs • - All PML variants are long-term unstable! • - SES3D monitors the total kinetic energy Etotal. • - When Etotal increases quickly, the PMLs are switched off and … • - … absorbing boundaries are replaced by less efficient multiplication by small numbers. • The poles and the core • - Elements become infinitesimally small at the poles and the core. • - SES3D is efficient only when the computational domain is sufficiently far from • the poles and the core. • Seismic discontinuities and the crust • - SEM is very accurate only when discontinuities coincide with element boundaries. • - SES3D‘s static grid may not always achieve this. • - It is up to the user to assess the numerical accuracy in cases where discontinuities run • through elements. [Implement long-wavelength equivalent models.] • - Generally no problem for the 410 km and 660 km discontinuities.

30. Thanks for your attention!